What river flows through the Old Forest?

The river that flows through the Old Forest is the Withywindle.

The Withywindle is a major river of the Old Forest in Tolkien's Middle-earth. It runs through the woodland's central valley, a region known as the Dingle, before continuing toward the Brandywine. The river's course is important because the forest's strange trees and paths tend to funnel unwary travellers toward it.

Frodo and his companions reach the Withywindle after becoming lost in the Old Forest. Near its banks they encounter Old Man Willow, an ancient willow tree with a malevolent influence. The tree traps the Hobbits until Tom Bombadil arrives and rescues them, making the river valley one of the most memorable episodes in The Fellowship of the Ring.

Withywindle is easily confused with the Brandywine, which borders the Shire and receives it as a tributary. Entwash belongs to Rohan, while Silverlode flows through the lands near Lothlórien. Thus the Withywindle is the specifically correct answer for the Old Forest.